Actually,

. You can see that it is simpler in radians. That limit is very important.

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Okay,

#4.

Note that,

, yes it is true to say the limit does not exist, but I wanted you to state that is keeps getting smaller which is what this shows.

And,

, yes it is true to say the limit does not exist, but I wanted you to state that is keeps getting larger which is what this shows.

#3 is confusing.

(In fact, I did not think this was going to be difficult).

Note youcannotapproach the limitfrom the leftbecause, you have a number less than 1 and when you take its reciprical it is larger than 1, and when from 1 you subtract this value (which is larger) you have a negative number. But you cannot have a negative number in the radical, thus it does not exist. Thus,

does not exist and it is not even one of those infinite limits.

But,

Because,

We can see the limit is zero.

Just to make you more hungry for limits, why is the limit 0? Because we say it approaches 0. But what about -1? It also approaches -1, surly. Because 0 is less than -1. In fact, 0 is the smallest number which it approaches and does not exceede. This fabolous result is calledWeierstrauss-Bolzano Theorem.

So, I hope you understand limits. What now? I can lecture about,

1)Conics

2)Mathematical Induction

3)More limits (this times we learn rule to compute limits).

...?